Since his introductory news conference in 2011, Todd Graham has talked about turning Arizona State into a national championship team.

The Sun Devils took steps toward that goal during his first two seasons as coach, winning a bowl game and capturing the 2013 Pac-12 South title.

Arizona State overcame another stumbling block this season, finally knocking off Stanford, the disciplined, character-driven program Graham wants his team to emulate.

The next hurdle was to win a high-stakes game on the national stage, something that had tripped up the Sun Devils in the past.

Arizona State finally solved that piece of the puzzle Saturday night in the desert, racing out to a huge lead against No. 8 Notre Dame then digging deep after the Irish put together a furious rally.

With their 55-31 win, the Sun Devils remained in the thick of the College Football Playoff race and proved to themselves — and everyone else — that they have the type of team that can win big games.

"We knew this was kind of the hump we needed to get over," said Arizona State running back D.J. Foster, who ran for 120 yards against the Irish.

Doubt had risen in the desert during Graham's first two seasons.

The Sun Devils had a good team. But the leap from good to great can be a big one and Arizona State had trouble navigating the road to greatness over those first two seasons.

In 2012, Graham's first year at the helm, the Sun Devils faced No. 2 Oregon at home in what was dubbed as one of the program's games in years. Arizona State fell flat and was run over by the Ducks 43-21.

Last season, Arizona State lost twice to Stanford. The second, in the Pac-12 Championship, cost the Sun Devils their first trip to the Rose Bowl since 1997.

Arizona State's big-game luck changed against the Irish, though it wasn't easy.

The Sun Devils raced out to a 31-point lead, fell back on their heels as Notre Dame scored 28 points, then summoned the will to close the game out, scoring three touchdowns in the final five minutes.

With its marquee victory, Arizona State moved up to No. 7 in The Associated Press poll and will likely climb a spot or two from No. 9 when the next College Football Playoff rankings are released on Tuesday.

"It's huge to get a win on a national stage like this," said Sun Devils quarterback Taylor Kelly, who threw for 224 yards and three touchdowns.

With the loss, Notre Dame fell off the playoff stage.

The Irish have two losses and no true marquee wins, so they're most likely looking at a New Year's Day bowl at best — still an accomplishment, but not what they had envisioned at the start of the season.

"There is no excuse. You can't point to anything," said Irish running back Cam McDaniel, who scored on a pair of 1-yard runs during Notre Dame's comeback. "It wasn't the heat. It wasn't the atmosphere. It wasn't anything. We know how to play in big games. We've won big games. There's no excuse why we didn't come back in the second half. We had em."

The Sun Devils now have their next goal in sight: a spot in college football's inaugural playoff.

Arizona State plays at Oregon State this week, Washington State at home after that and closes out the regular season against rival Arizona in Tucson.

Get through that stretch without a loss, the Sun Devils will face their next big test, likely against No. 3 Oregon in the Pac-12 Championship game.

Win that and it would all but lock Arizona State into the playoff.

"All the pieces of the puzzle are coming together," Foster said. "It's a sight to see."